Tag: Newton’s Law of Gravity

A beautiful April day, far too nice to be inside working. I’m on a brisk walk toward the lake when I hear puffing behind me. “Hey, Moire, I got questions!”

“Of course you do, Mr Feder. Ask away while we hike over to watch the geese.”

“Sure, but slow down , will ya? I been reading this guy’s blog and he says some things I wanna check on.”

I know better but I ask anyhow. “Like what?”

“Like maybe the planets have different electrical charges so if we sent an astronaut they’d get killed by a ginormous lightning flash.”

“That’s unlikely for so many reasons, Mr Feder. First, it’d be almost impossible for the Solar System to get built that way. Next, it couldn’t stay that way if it had been. Third, we know it’s not that way now.”

“One at a time.”

“OK. We’re pretty sure that the Solar System started as a kink in a whirling cloud of galactic dust. Gravity spanning the kink pulled that cloud into a swirling disk, then the swirls condensed to form planets. Suppose dust particles in one of those swirls, for whatever reason, all had the same unbalanced electrical charge.”

“Right, and they came together because of gravity like you say.”

I pull Old Reliable from its holster. “Think about just two particles, attracted to each other by gravity but repelled by their static charge. Let’s see which force would win. Typical interstellar dust particles run about 100 nanometers across. We’re thinking planets so our particles are silicate. Old Reliable says they’d weigh about 2×10 –18 kg each, so the force of gravity pulling them together would be … oh, wait, that’d depend on how far apart they are. But so would the electrostatic force, so let’s keep going. How much charge do you want to put on each particle?”

“The minimum, one electron’s worth.”

“Loading the dice for gravity, aren’t you? Only one extra electron per, umm, 22 million silicon atoms. OK, one electron it is … Take a look at Old Reliable’s calculation. Those two electrons push their dust grains apart almost a quintillion times more strongly than gravity pulls them together. And the distance makes no difference — close together or far apart, push wins. You can’t use gravity to build a planet from charged particles.”

“Sure, which is why I said almost impossible. Now for the second reason the astronaut won’t get lightning-shocked — the solar wind. It’s been with us since the Sun lit up and it’s loaded with both positive- and negative-charged particles. Suppose Venus, for instance, had been dealt more than its share of electrons back in the day. Its net-negative charge would attract the wind’s protons and alpha particles to neutralize the charge imbalance. By the same physics, a net-positive planet would attract electrons. After a billion years of that, no problem.”

“All right, what’s the third reason?”

“Simple. We’ve already sent out orbiters to all the planets. Descent vehicles have made physical contact with many of them. No lightning flashes, no fried electronics. Blows my mind that our Cassini mission to Saturn did seven years of science there after a six-year flight, and everything worked perfectly with no side-trips to the shop. Our astronauts can skip worrying about high-voltage landings.”

“Hey, I just noticed something. Those F formulas look the same.” He picks up a stick and starts scribbling on the dirt in front of us. “You could add them up like F=(Gm1m2+k0q1q2)/r2. See how the two pieces can trade off if you take away some mass but add back some charge? How do we know we’ve got the mass-mass pull right and not mixed in with some charge-charge push?”

“Good question. If protons were more positive than electrons, electrostatic repulsion would always be proportional to mass. We couldn’t separate that force from gravity. Physicists have separately measured electron and proton charge. They’re equal (except for sign) to 10 decimal places. Unfortunately, we’d need another 25 digits of accuracy before we could test your hypothesis.”

If you’ve ever watched or read a space opera (oh yes, you have), you know about the gravity well that a spacecraft has to climb out of when leaving a planet. Every time I see the Museum’s gravity well model (photo below), I’m reminded of all the answers the guy gave to, “Johnny, what can you make of this?”

The model’s a great visitor-attracter with those “planets” whizzing around the “Sun,” but this one exhibit really represents several distinct concepts. For some of them it’s not quite the right shape.

The simplest concept is geometrical. “Down” is the direction you move when gravity’s pulling on you.

Gravitational potential energy changefor small height differences

A gravity well model for that concept would be just a straight line between you and the neighborhood’s most intense gravity source.

You learned the second concept in high school physics class. Any object has gravitational potential energy that measures the amount of energy it would give up on falling. Your teacher probably showed you the equation GPE = m·g·h, where m is the mass of the object, h is its height above ground level, and g is a constant you may have determined in a lab experiment.

If the width of the gravity well model at a given height represents GPE at that level, the model is a simple straight-sided cone.

But of course it’s not that simple. Newton’s Law of Gravity says that the potential energy at any height r away from the planet’s center is proportional to 1/r.

Hmm… that looks different from the “proportional to h” equation. Which is right?

Both equations are valid, but over different distance scales. The HS teachers didn’t quite lie to you, but they didn’t give you the complete picture either. Your classroom was about 4000 miles (21,120,000 feet) from Earth’s center, whereas the usual experiments involve height differences of at most a dozen feet. Even the 20-foot drop from a second-story window is less than a millionth of the way down to Earth’s center.

Check my numbers:

Height h

1/(r+h)
× 108

Difference in 1/(r+h)
× 1014

0

4.734,848,484

0

20

4.734,844,001

4.48

40

4.734,839,517

8.97

60

4.734,835,033

13.45

80

4.734,830,549

17.93

100

4.734,826,066

22.42

Sure enough, that’s a straight line (see the chart). Reminds me of how Newton’s Law of Gravity is valid except at very short distances. The HS Law of Gravity works fine for small spans but when the distances get big we have to use Newton’s equation.

We’re not done yet. That curvy funnel-shaped gravity well model could represent the force of gravity rather than its potential energy. Newton told us that the force goes as 1/r2 so it decreases much more rapidly than the potential energy does as you get further away. The gravity force well has a correspondingly sharper curve to it than the gravity energy well.

The force of gravityor an embedding diagram

The funnel model could also represent the total energy required to get a real spacecraft off the surface and up into space. Depending on which sci-fi gimmickry is in play, the energy may come from a chemical or ion rocket, an electromagnetic railgun, or even a tractor beam from some mothership way up there.

No matter the technology, the theoretical energy requirement to get to a given height is the same. In practice, however, each technology is optimal for some situations but forbiddingly inefficient in others. Thus, each technology’s funnel has its own shape and that shape will change depending on the setting.

In modern physics, the funnel model could also represent Einstein’s theory of how a mass “bends” the space around it. (Take a look at this post, which is about how mass curves space by changing the local distance scale.) Cosmologists describe the resulting “shapes” with embedding diagrams that are essentially 2D pictures of 3D (or 4D) contour plots. The contours are closest together where space is most compressed, just as lines showing a steep hillside on a landscape contour map are close together.

The ED around a non-spinning object looks just like the force model picture above. No surprise — gravitational force is how we we perceive spatial curvature.

~~ Rich Olcott

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